Remark

Properties

The properties of the category of monoids Mon(C)Mon (C), especially with respect to colimits, are markedly different according to whether or not the tensor product of CC preserves colimits in each variable. (This is automatically the case if CC is closed.)

Most “algebraic” situations have this property, but others do not. For instance, the category of monads on a fixed category AA is Mon(C)Mon (C), where C=[A,A]C= [A,A] is the category of endofunctors of AA with composition as its monoidal structure. This monoidal product preserves colimits in one variable (since colimits in [A,A][A,A] are computed pointwise), but not in the other (since most endofunctors do not preserve colimits). So far, the material on this page focuses on the case where ⊗\otimes does preserve colimits in both variables, although some of the references at the end discuss the more general case.

in CC, with the monoidal structure given by tensor product/juxtaposition.

Proof

A morphism f:FCX→Af : F_C X \to A in Mon(C)Mon(C) with components fk:X⊗k→UCAf_k : X^{\otimes k} \to U_C A is entirely fixed by its component f˜=f1:X→UCA\tilde f = f_1 : X \to U_C A on XX, because by the homomorphism property and the special free nature of the product in FCXF_C X

Of commutative monoids

Proposition

that are preserved by the tensor product functors A⊗(−):𝒞→𝒞A \otimes (-) \colon \mathcal{C} \to \mathcal{C} for all objects AA in 𝒞\mathcal{C}.

Then for f:A→Bf \colon A \to B and g:A→Cg \colon A \to C two morphisms in the category CMon(\mathcal{}) of commutative monoids in 𝒞CMon(\mathcal{}\mathcal{C}, the underlying object in 𝒞\mathcal{C} of the pushout in CMon(𝒞)CMon(\mathcal{C}) coincides with that of the pushout in the category AAMod of AA-modules

U(B∐AC)≃B⊗AC.
U(B \coprod_A C) \simeq B \otimes_A C
\,.

Here BB and CC are regarded as equipped with the canonical AA-module structure induced by the morphisms ff and gg, respectively.

of objects (Pn)n∈ℕ(P_n)_{n \in \mathbb{N}}, which are each given by pushouts in CC inductively as follows.

Assume Pn−1P_{n-1} has been defined. Write Sub(n)Sub(\mathbf{n}) for the poset of subsets of the nn-element set n\mathbf{n} (this is the poset of paths along the edges of an nn-dimensional cube). Define a diagram

where the top morphism is the canonical one induced by the commutativity of the diagram KK, and where the left morphism is defined in terms of components K−(S)K^-(S) of the colimit for S⊂nS \subset \mathbf{n} a proper subset by the tensor product morphisms of the form

This gives the underlying object of the monoid PP. Take the monoid structure on it as follows. The unit of PP is the composite

eP:IC→eXX→P
e_P : I_C \stackrel{e_X}{\to} X \to P

with the unit of XX. The product we take to be the image in the colimit of compatible morphisms Pk⊗Pk→Pk+lP_k \otimes P_k \to P_{k + l} defined by induction on lk+llk + l as follows. we observe that we have a pushout diagram

where Qn:=(lim→K)nQ_n := (\lim_{\to} K)_n is the colimit as in the above at stage nn.

There is a morphism from the bottom left object to Pk+lP_{k+l} given by the induction assumption. Moreover we have a morphism from the top right object to Pk+1P_{k+1} obtained by first multiplying the two adjacent factors of XX and then applying the morphism (X⊗L)⊗k+l⊗X→Pk+l(X \otimes L)^{\otimes k+l} \otimes X \to P_{k+l}. These are compatible and hence give the desired morphism Pk⊗Pk→Pk+lP_k \otimes P_k \to P_{k+l}.